Simplify the following expression and state the condition under which the simplification is valid. $n = \dfrac{-10p^2 - 30p - 20}{-2p^3 + 16p^2 + 40p}$
First factor out the greatest common factors in the numerator and in the denominator. $ n = \dfrac {-10(p^2 + 3p + 2)} {-2p(p^2 - 8p - 20)} $ $ n = \dfrac{10}{2p} \cdot \dfrac{p^2 + 3p + 2}{p^2 - 8p - 20} $ Simplify: $ n = \dfrac{5}{p} \cdot \dfrac{p^2 + 3p + 2}{p^2 - 8p - 20}$ Next factor the numerator and denominator. $ n = \dfrac{5}{p} \cdot \dfrac{(p + 2)(p + 1)}{(p + 2)(p - 10)}$ Assuming $p \neq -2$ , we can cancel the $p + 2$ $ n = \dfrac{5}{p} \cdot \dfrac{p + 1}{p - 10}$ Therefore: $ n = \dfrac{ 5(p + 1)}{ p(p - 10)}$, $p \neq -2$